An Efficient and Long-Time Accurate Third-Order Algorithm for the Stokes–Darcy System

A third order in time numerical IMEX-type algorithm for the Stokes–Darcy system for flows in fluid saturated karst aquifers is proposed and analyzed. a novel third-order Adams–Moulton scheme is used for the discretization of the dissipative term whereas a third-order explicit Adams–Bashforth scheme is used for the time discretization of the interface term that couples the Stokes and Darcy components. the scheme is efficient in the sense that one needs to solve, at each time step, decoupled Stokes and Darcy problems. Therefore, legacy Stokes and Darcy solvers can be applied in parallel. the scheme is also unconditionally stable and, with a mild time-step restriction, long-time accurate in the sense that the error is bounded uniformly in time. Numerical experiments are used to illustrate the theoretical results. to the authors\u27 knowledge, the novel algorithm is the first third-order accurate numerical scheme for the Stokes–Darcy system possessing its favorable efficiency, stability, and accuracy properties

A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation

We propose a novel second order in time numerical scheme for Cahn-Hilliard-Navier- Stokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, we prove that the scheme is uncondition- ally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the weak coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme

Decoupling methods for the time-dependent Navier-Stokes-Darcy interface model

In this research, several decoupling methods are developed and analyzed for approximating the solution of time-dependent Navier-Stokes-Darcy (NS-Darcy) interface problems. This research on decoupling methods is motivated to efficiently solve the complex Stokes-Darcy or NS-Darcy type models, which arise from many interesting real world problems involved with or even dominated by the coupled porous media flow and free flow. We first discuss a semi-implicit, multi-step non-iterative domain decomposition (NIDDM) to solve a coupled unsteady NS-Darcy system with Beavers-Joseph-Saffman-Jones (BJSJ) interface condition and obtain optimal error estimates. Second, a parallel NIDDM is developed to solve unsteady NS-Darcy model with Beavers-Joseph (BJ) interface condition, which is much more complicated than BJSJ interface condition. We overcome the major difficulties in the analysis which arise from nonlinear terms and BJ interface condition. Furthermore, a Lagrange multiplier method is proposed under the framework of the domain decomposition method to overcome the difficulty of non-unique solutions arising from the defective boundary condition. Meanwhile, we propose and analyze an efficient ensemble algorithm, which can significantly improve the computational efficiency, for fast computation of multiple realizations of the stochastic Stokes-Darcy model with a random hydraulic conductivity tensor. Furthermore, we utilize the idea of artificial compressibility, which decouples the velocity and pressure, to construct the decoupled ensemble algorithm to improve computational efficiency further. We prove that the proposed ensemble methods offer long time stability and optimal error estimates under a time-step condition and two parameter conditions --Abstract, page iii

Efficient and Long-Time Accurate Second-Order Methods for the Stokes-Darcy System

We propose and study two second order in time implicit-explicit methods for the coupled Stokes-Darcy system that governs flows in karst aquifers and other subsurface flow systems. the first method is a combination of a second-order backward differentiation formula and the second order Gear\u27s extrapolation approach. the second is a combination of the second-order Adams-Moulton and second-order Adams-Bashforth methods. Both algorithms only require the solution of decoupled Stokes and Darcy problems at each time-step. Hence, these schemes are very efficient and can be easily implemented using legacy codes. We establish the unconditional and uniform in time stability for both schemes. the uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes, e.g., contaminant sequestration and release, that occur over very long-time scales. Error estimates for fully discretized schemes using finite element spatial discretization\u27s are derived. Numerical examples are provided that illustrate the accuracy, efficiency, and long-time stability of the two schemes. © 2013 Society for Industrial and Applied Mathematics

A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system

A multilevel Monte Carlo (MLMC) method for Uncertainty Quantification (UQ) of advection-dominated contaminant transport in a coupled Darcy–Stokes flow system is described. In particular, we focus on high-dimensional epistemic uncertainty due to an unknown permeability field in the Darcy domain that is modelled as a lognormal random field. This paper explores different numerical strategies for the subproblems and suggests an optimal combination for the MLMC estimator. We propose a specific monolithic multigrid algorithm to efficiently solve the steady-state Darcy–Stokes flow with a highly heterogeneous diffusion coefficient. Furthermore, we describe an Alternating Direction Implicit (ADI) based time-stepping for the flux-limited quadratic upwinding discretization for the transport problem. Numerical experiments illustrating the multigrid convergence and cost of the MLMC estimator with respect to the smoothness of permeability field are presented